We prove that in our case, all the above containments hold with equality. For every \(i\in [n]\) , let \(K_i\)  denote the 0-dimensional abstract simplicial complex consisting of the 1-element subsets of \(X_i\)  and the empty set. Consider now their join \(K=K_1\ast \ldots \ast K_n\) ; see [1]}, Chapter 0. By definition, \(K\)  is the same as the simplicial complex \(\cup _{i=0}^nS_i\) .
